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 A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x. 16
 1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1, 1, 1014 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215. LINKS Alois P. Heinz, Rows n = 0..140, flattened Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22. FORMULA E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001 T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018 EXAMPLE Triangle begins: 1; 1, 1; 1, 2, 1; 1, 5, 5, 1; 1, 12, 22, 12, 1; 1, 27, 92, 92, 27, 1; 1, 58, 359, 604, 359, 58, 1; 1, 121, 1311, 3607, 3607, 1311, 121, 1; ... MATHEMATICA t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k]; A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k]; T[n_, k_] := A[n, k] + A[n, k+1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, after Franck Maminirina Ramaharo *) CROSSREFS Cf. A008292. Columns include A000325. Row sums are in A098558. T(2n,n) gives A177042. Sequence in context: A196636 A196641 A197090 * A264862 A176420 A099927 Adjacent sequences: A008515 A008516 A008517 * A008519 A008520 A008521 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane. EXTENSIONS More terms from Vladeta Jovovic, Apr 06 2001 STATUS approved

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Last modified May 27 20:09 EDT 2023. Contains 362985 sequences. (Running on oeis4.)